A class of generalized B-spline curves

The classical B-spline functions of order $k geq 2$ are recursively defined as a special combination of two consecutive B-spline functions of order $k-1$. At each step, this recursive definition is based, in general, on different reparametrizations of the strictly increasing identity (linear core) function $varphi\left(u\right)=u$. This paper generalizes the concept of the classical normalized B-spline functions by considering monotone increasing continuously differentiable nonlinear core functions instead of the classical linear one. These nonlinear core functions are not only interesting from a theoretical perspective, but they also provide a large variety of shapes.

We show that many advantageous properties (like the non-negativity, local support, the partition of unity, the effect of multiple knot values, the special case of Bernstein polynomials and endpoint interpolation conditions) of the classical normalized B-spline functions remain also valid for this generalized case, moreover we also provide characterization theorems for not so obvious (geometrical) properties like the first and higher order continuity of the generalized normalized B-spline functions, $C^1$ continuous envelope contact property of the family of curves obtained by altering a selected knot value between its neighboring knots. Characterization theorems are illustrated by test examples.

We also outline new research directions by ending our paper with a list of open problems and conjectures underpinned by numerous successful numerical tests.